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Adaptive control of time-delay systems to counteract pathological brain oscillations

Authors
  • Orłowski, Jakub
Publication Date
Dec 19, 2019
Source
Kaleidoscope Open Archive
Keywords
Language
English
License
Unknown
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Abstract

Beta oscillations (10-30 Hz) observed in the basal ganglia are a well-known biomarker of Parkinson's disease, correlated with increased symptoms of akinesia and bradykinesia. Deep brain stimulation (DBS) leads to a reduction of these oscillations, as well as improvement in the patients' quality of life. Clinically used DBS, however, is since its inception delivered in an open-loop fashion, where the parameters of the stimulation are constant regardless of the underlying brain activity and the state of the patient. This can lead to overstimulation, inducing side-effects and shortening battery life of the impulse generator, as well as understimulation when the symptoms of the disease worsen. Closed-loop DBS, exploiting measurements on the patient's brain activity to adapt the stimulation in real-time, is a promising way to overcome these limitations. In this thesis, we rely on an existing firing-rate model of the activity of the subthalamic nucleus (STN) - external globus pallidus (GPe) loop to propose an adaptive proportional DBS.We first analyze the model under proportional feedback and show that high-gain proportional stimulation makes the system globally exponentially stable (GES). To that aim, we propose a relaxed Lyapunov-Krasovskii condition for GES, valid for globally Lipschitz systems. We then extend the sigma modification approach, originally proposed by Ioannou and Kokotovic, to time-delay systems by providing explicit conditions under which this adaptive control stabilizes the system. We show that this controller then induces a practical stability property, in which the L_1 norm of the state over a sufficiently long time window converges to a neighborhood of the equilibrium up to a steady-state error that can be made arbitrarily small by tuning a control parameter. When applied to the STN-GPe firing-rate model, this leads to a proportional control law, whose gain is automatically adjusted based on the measured activity of the STN, to successfully disrupt pathological brain oscillations. In an attempt to assess the robustness of this adaptive control strategy to exogenous inputs or unmodeled dynamics, we also disprove an existing result on partial stability of nonlinear systems.Finally, we illustrate with numerical simulations on a spatiotemporal extension of this model that the proposed control law is capable of selectively quenching the pathological oscillations, based on their frequency band, regardles of whether the oscillations originate within the STN-GPe loop, or in the cortical neurons projecting to the STN.

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